Communication transmission with super-gaussian filtering in receiver

ABSTRACT

In a transmission system, a transmission signal is generated from transmission data. The transmission signal has a series of shaped pulses, such as raised-cosine or root-raised-cosine shaped pulses, formed using a pulse-shaping filter. A reception signal, based on the transmission signal having passed through a transmission channel, is sampled to generate sampled digital data. The sampled digital data is filtered through a receiver filter to regenerate the transmission data, where the pulse shaping filter and the receiver filter are mismatched.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/882,862 filed on Jan. 29, 2018, and entitled “COMMUNICATIONTRANSMISSION WITH SUPER-GAUSSIAN FILTERING IN RECEIVER”; which claimsthe benefit of U.S. Provisional Patent Application No. 62/453,196 filedon Feb. 1, 2017, and entitled “COMMUNICATION TRANSMISSION WITHSUPER-GAUSSIAN FILTERING IN RECEIVER;” which is hereby incorporated byreference for all purposes.

BACKGROUND OF THE INVENTION

In the electrical or optical transmission of digital information bydigital modulation, the transmitter typically encodes the data bits ofthe digital information and filters sequences of the data bits to formpulses for transmission through a channel, and the receiver typicallytakes digital samples of the received pulses and filters the digitalsamples to retrieve or regenerate the original digital information. Asunambiguously asserted by theory, the information-bearing waveforms usedfor transmission of information in communication systems ought tosatisfy the so called Nyquist criterion so as to avail communicationwithout intersymbol interference (ISI). In practice, the family ofwaveforms used almost exclusively are the raised-cosine and root-raisedcosine pulses, often interchangeably, although strictly speakingerroneously denoted as the Nyquist pulses. The ideal Nyquist pulseswould be infinite in length. However, in practical implementations,i.e., in real systems, the pulses have to be truncated at thetransmitter and sampled at a finite number of points or intervals at thereceiver. To do this, the filter functions in the transmitter andreceiver use filters referred to as finite impulse response (FIR)filters. The finite aspect of the filters both affects the spectralshape of the transmitted pulses and the effectiveness of the subsequentfiltering response by the receiver. The result of the responsetruncation is departure from ideal Nyquist pulses and generallyappearance of the ISI, which refers to distortions of a transmissionsignal in which one symbol (represented by a pulse) interferes withother adjacent or nearby symbols. In addition to the intended pulseshaping, a signal usually picks up various impairments, distortions andnoise when passing through the transmission channel.

FIGS. 1 and 2 illustrate the general effect of ISI. FIG. 1 shows anexample graph 101 of transmission pulses (for two bits 1,0) produced bya transmitter and a graph 102 of reception pulses received by areceiver. The transmission pulses in this example are idealized assquare waves, but in reality the vertical edges (i.e., edges of thesignal band) have a finite slope. The reception pulses, on the otherhand, tend to get elongated and smeared out. As long as the receptionpulses do not overlap, there is no ISI. Thus, the receiver can samplethe reception signal at any point within the same pulse intervals of theoriginal transmission pulses (as indicated by dashed lines) and producethe correct data. However, FIG. 2 shows an example graph 201 oftransmission pulses (for five bits 1,0,1,1,0) produced by a transmitter,a graph 202 of the reception pulses for each transmission pulse, and agraph 203 of a reception signal received by a receiver. In this case,the elongated and smeared out reception pulses overlap, so that the neteffect detected by the receiver is the irregular reception signal ofgraph 203. Thus, as long as the receiver samples the reception signal atproper locations, e.g., as indicated by the dots, then the correct datawill be obtained. However, if the receiver samples the reception signalwithin the subintervals indicated by arrows 204-206, the receiver willobtain incorrect data, even though the sample locations would be withinthe correct pulse intervals of the original transmission pulses. Thepotential for generating incorrect data is the overall issue that mustbe avoided or minimized.

ISI can be caused by many different reasons. For example, it can becaused by filtering effects from hardware or frequency selective fading,multipath interference, from non-linearities, and fromcharging/discharging effects. Very few systems are immune to ISI, so itis nearly always present in communication systems. Thus, communicationsystem designs nearly always need to incorporate some way ofcontrolling, mitigating or minimizing ISI.

One of the simplest solutions for reducing ISI is to simply slow downthe transmission rate of the signal that is passed through the channel,e.g., with a delay between each pulse as illustrated by FIG. 1. Thus,the next pulse of information is transmitted only after allowing thecurrent received pulse to damp down, so that the subsequent pulse doesnot interfere with the current pulse. Slowing down the transmissionrate, however, is an easy, but unacceptable, solution. Instead, it isdesired to be able to transmit the pulses at a much higher rate, asillustrated by FIG. 2.

To provide the best transmission rate through the channel, ISI generallyhas to be minimized without providing a delay between transmissionpulses. To be able to handle the higher transmission rate, the primarytechniques used to counter ISI involve “pulse-shaping.” Pulse-shapingtechniques generally modulate the pulses with a particular shape at thetransmitter, and use digital demodulation processes at the receiver, insuch a manner that the points at which the pulses are sampled are onlyminimally affected by interference.

The square (or almost square) pulse shapes in the examples of FIGS. 1and 2 are generally inadequate for pulse shaping purposes, as theycannot be accomplished in practice. Instead, the pulse shaping iscommonly based on other forms, such as a sinc pulse, a raised-cosine(RC) (or root-raised-cosine (RRC)) pulse, or a Nyquist pulse, asillustrated by an ideal RC/RRC time domain pulse 301 in FIG. 3. Theoscillations in the left and right side tails of the time domain pulse301 are slowly diminished, but never truly die out, which isillustrative of the infinite length of the ideal pulse mentioned above.The dots, on the other hand, represent sampling locations (or tappoints) used for finite generation (synthesis) and analysis of thepulses. Any number of taps can be used, e.g., 16, 32, 48, 64, 128.Generally, a larger number of more tightly spaced tap points provides ahigher quality result. However, the larger number of tap points alsogenerally requires more complicated, or costly hardware and/or higherpower dissipation of the associated hardware.

To minimize ISI, therefore, the transmitter and receiver commonly useraised-cosine filters or root-raised-cosine filters to shape the pulsesat the transmitter and handle the response at the receiver. Variationson each type of filter, however, result in pulses with different shapes.In order to get the best results for minimizing ISI, therefore, it isgenerally accepted that the filter in the receiver must match the filterin the transmitter. To implement the raised-cosine response, forexample, the filtering is split into two parts to create a matched set.When the raised-cosine filtering is split into two parts, each part iscalled the root-raised-cosine.

SUMMARY OF THE INVENTION

In some embodiments, a method involves a transmitter receivingtransmission data, generating a transmission signal having a series ofshaped pulses (e.g., a series of raised-cosine or root-raised-cosineshaped pulses) from the transmission data, and transmitting thetransmission signal. The method further involves a receiver receiving areception signal based on the transmission signal having passed througha transmission channel, sampling the reception signal to generatesampled digital data, and filtering the sampled digital data through asuper-Gaussian filter to regenerate the transmission data.

In some embodiments, a transmission system includes a transmitter and areceiver. The transmitter has a pulse shaping filter (e.g., araised-cosine or root-raised-cosine filter) with which the transmittergenerates a transmission signal with a series of shaped pulses fortransmission. The series of shaped pulses are generated from digitaltransmission data. The receiver has a super-Gaussian filter with whichthe receiver regenerates the digital transmission data from sampleddigital data of a reception signal based on the transmission signalhaving passed through a transmission channel.

In some embodiments, a method involves a receiver receiving a receptionsignal based on a transmission signal having passed through atransmission channel, the transmission signal having been formed with aseries of shaped pulses (e.g., raised-cosine or root-raised-cosineshaped pulses), and the transmission signal having been formed fromdigital transmission data. The method further involves the receiversampling the reception signal to generate sampled digital data, andfiltering the sampled digital data through a super-Gaussian filter toregenerate the digital transmission data.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2 show simplified graphs of transmission and receptionpulses.

FIG. 3 shows a simplified graph of an ideal time domain pulse.

FIG. 4 is a simplified schematic of a transmission system in accordancewith some embodiments.

FIG. 5 shows a simplified ideal RC/RRC frequency response graph.

FIG. 6 shows a simplified ideal super-Gaussian frequency response graph.

FIG. 7 shows simplified graphs of Q-factor vs. signal-to-noise ratio(SNR) for several example transmission systems.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Reference now will be made in detail to embodiments of the disclosedinvention, one or more examples of which are illustrated in theaccompanying drawings. Each example is provided by way of explanation ofthe present technology, not as a limitation of the present technology.In fact, it will be apparent to those skilled in the art thatmodifications and variations can be made in the present technologywithout departing from the scope thereof. For instance, featuresillustrated or described as part of one embodiment may be used withanother embodiment to yield a still further embodiment. Thus, it isintended that the present subject matter covers all such modificationsand variations within the scope of the appended claims and theirequivalents.

The systems and methods described herein can be used in communicationsystems that utilize different modulation schemes, signal processingfunctionalities, and/or signal processing capabilities, including butnot limited to quadrature amplitude modulation (QAM), orthogonalfrequency-division multiplexing (OFDM), code-division multiple access(CDMA), RAKE receiver beam combining, and phased-array basedtransmission and reception. The communication systems can includemultiple functionalities and/or processing capabilities to addressmultiple types of impairments, for example multipath impairments.Communication systems using linear modulation and/or signal processingtechniques can benefit from the super-Gaussian filter described herein.Furthermore, the super-Gaussian filter can be applied anywhere in theprocessing chain for such linear systems.

Signal shaping at the receiver is accomplished by filtering, which alsoserves the functionality of rejecting any out-of-band noise that arriveswith the useful signal at the receiving end of the system. The shapingat the transmitter can be combined with some other functionalities.Conventional communication systems typically employ filters in thereceiver that are matched with the signal shape. In other words, thesame types of filters are used to filter the signal in the transmitter(before transmitting through a channel) and in the receiver (to filterthe received signal after transmission through the channel). In theory,matched filters in the transmitter and receiver are ideal. However, inpractice, pulse shaping filters and/or imperfect components in thesystem chain, such as raised-cosine (RC) or root-raised-cosine (RRC)filters, introduce distortions in the signal pulses, which are notperfectly corrected for using a matched filter in the receiver.Described herein are communication systems with mismatched filters inthe transmitters and receivers, which provide the unexpected result ofbetter performance than can be achieved with assumed ideal matchedfilters.

In some embodiments, the filters used in the receivers described hereincan have excess bandwidth compared to the filters used in thetransmitters. For example, in some embodiments, a transmitter has apulse shaping filter in the RC or RRC filter families to shape thepulses, and a receiver has a super-Gaussian (SG) filter acting in thedigital domain, where the bandwidth of the SG filter is larger than thebandwidth of the pulse shaping filter in the transmitter. In someembodiments, the excess bandwidth in the SG filter in the receiver is atleast as large as the roll-off-factor in the RC or RRC filter in thetransmitter. In other words, the 3 dB bandwidth of the SG filter in thereceiver can be related to the RC or RRC bandwidth by

$\begin{matrix}{{f_{3\; {dB}} = {\frac{R}{2}\left( {1 + \beta} \right)C}},} & (1)\end{matrix}$

where β is the roll-off-factor of the RC or RRC filter in thetransmitter, R is the symbol rate of the signal, and C is a coefficientbetween 1 and 2.

FIG. 4 shows a transmission system 400 with a transmitter 401 and areceiver 402 for transmitting a transmission signal (e.g., electrical oroptical) through a transmission channel 403 (e.g., electrical cable,optical fiber, air, etc.), in accordance with some embodiments. In someembodiments, the electrical transmission system 400 is an RFtransmission system, transmitting an RF signal, and the transmissionchannel 403 is air. In some embodiments, an optical channel 403 in anoptical transmission system 400 is a fiber-optic cable, or is free spacethrough which an optical signal is transmitted. Contrary to generallyaccepted practices mentioned above, the transmission system 400 does notuse a matched set of filters in the transmitter 401 and the receiver402. Instead, in some embodiments, whereas a filter 404 in thetransmitter 401 uses a raised-cosine (RC) or root-raised-cosine (RRC)function to shape transmission pulses for a waveform of the transmissionsignal, a filter 405 in the receiver 402 uses a super-Gaussian functionto handle the response for analyzing the waveform of the receptionsignal. This mismatch in transmitter and receiver filtering functionshas been found to result in a markedly improved performance in handlingthe response at the receiver 402, while even using fewer sampling pointsto analyze the reception pulses. In some embodiments, the sampling inthe receiver can be done at many different sampling rates. For example,the sampling in the receiver can be done at 2 samples per symbol, lessthan 2 samples per symbol, greater than 2 samples per symbol, or from 1to 10 samples per symbol. In particular, simulations and practicalimplementation have shown that with finite arithmetic and truncatedvariant of RC/RRC pulses, a markedly improved performance is obtained byimplementing the receiver filter 405 as a (discrete time or sampledvariant) super-Gaussian filter. In the frequency domain Raised Cosine:

$\begin{matrix}{{H(f)} = \left\{ {\begin{matrix}{1,} & {{f} \leq \frac{1 - \beta}{2\; T_{s}}} \\{\frac{1}{2}\left\lbrack {1 + {\cos \left( {\frac{\pi \; T_{s}}{\beta}\left\lbrack {{f} - \frac{1 - \beta}{2\; T_{s}}} \right\rbrack} \right)}} \right\rbrack} & {\frac{1 - \beta}{2\; T_{s}} < {f} \leq \frac{1 + \beta}{2\; T_{s}}} \\0 & {otherwise}\end{matrix},} \right.} & (2)\end{matrix}$

where β is the raised cosine roll-off (or the excess bandwidth)parameter, T_(s) is the symbol duration interval, f is frequency.

In addition to the transmitter filter 404, the transmitter 401 generallyincludes a source 406 for the input of an information stream, anerror-correcting code (ECC) encoder 407, an optional channel pre-coder408, an optional symbol mapper 409, and an arbitrary waveform generator410, among other components not shown for simplicity. Additionally, insome embodiments, some of the functions described for the components 404and 406-410 are implemented in other components, either shown or notshown.

The information source 406 generally represents circuitry (e.g., of anoverall electronic device of which the transmitter 401 is a portion)that generates the data (bits/bytes) for the information stream,including a message body that is to be transmitted to the receiver 402and headers designating the source and destination of the message. TheECC encoder 407 generally represents circuitry that inserts errorcorrection bits (to make the transmission data resilient to informationloss, noise or shortcomings of the transmission channel 403) into thedata for the information stream in order to generate the bits of thetransmission data. In some embodiments, the optional channel pre-coder408 represents circuitry that adds additional encoding bits into thetransmission data or performs an inversion of some bits of thetransmission data in order to further protect the data if desired. Insome embodiments, the optional symbol mapper 409 generally representscircuitry that maps multi-bit combinations of the bits of thetransmission data into multiple symbols (corresponding to multipleamplitude levels) for generating the transmission signal with more thantwo (e.g., 4, 8, 16, etc.) signal amplitude levels. At this point, theoutput of the components 407-409 represents the serialized bits (e.g.,the ones and zeroes, or the multi-level symbols) of the transmissiondata that are used to form the shaped pulses (or, information-bearingpulses) of the transmission signal.

The transmitter filter 404 generally represents digital pulse shapingfilter circuitry that performs the RC/RRC filtering of the serializedbits of the transmission data to generate a digital representation ofthe desired shaped pulses for the waveform of the transmission signal.The digital representation indicates the amplitude of the shaped pulsesat discrete points of an idealized RC/RRC pulse, e.g., as indicated bythe dots on the time domain pulse 301 (FIG. 3). The number and spacingof the dots are provided for illustrative and explanatory purposes only.The transmitter filter 404, thus, produces the digital representation ofthe shaped pulses with any appropriate number and spacing of discretepoints that is adequate for generating the desired shaped pulses. Thearbitrary waveform generator 410 (or digital channel converter,digital-to-analog converter, etc.) generally represents circuitry thatconverts the digital representation into the analog shaped pulses withan appropriate timing and with an amplitude indicated by the digitalrepresentation. The analog shaped pulses (i.e., the transmission signal)are then transmitted through the transmission channel 403. Otherembodiments can generate the transmission signal from the transmissiondata with a different implementation for, or variations in the shape of,the RC/RRC pulse.

Whereas the bits of the transmission data are mathematical constructs ofones and zeroes, the RC/RRC filtering into the digital and analogrepresentations and the generation of the transmission signal therefromprovide a real-world shape for the data in the form of the shapedpulses. The ones and zeroes of the original data are tantamount to anideal square wave, and the shaped pulses generally approximate an idealRC/RRC impulse response (in the time domain) for an ideal square wave(in the frequency domain).

In addition to the receiver filter 405, the receiver 402 generallyincludes an optional receiver and response shaping/adjustment filter411, a sampler 412, an optional channel impairments mitigating equalizerand data-retrieval filter 413, a de-mapper 414, a decoder 415, and anoutput 416 for the retrieved information stream, among other componentsnot shown for simplicity. Additionally, in some embodiments, some of thefunctions described for the components 405 and 411-416 are implementedin other components, either shown or not shown.

The optional receiver and response shaping/adjustment filter 411generally represents circuitry for any appropriate analog filter thatmay be used to receive the reception signal and initially filter out orreject a portion of the accumulated interference or noise in thereception signal. With better digital filtering, however, the optionalreceiver and response shaping/adjustment filter 411 may not be needed.

The sampler 412 generally represents circuitry (e.g., including ananalog-to-digital converter (ADC)) for sampling (i.e., capturing thesignal in discrete time instances, as opposed to a continuous manner,and with finite amplitude resolution—at those captured time instances)the reception signal at appropriate timing spaces or intervals, i.e., atthe sampling points, sufficient to adequately describe the waveform.Thus, the sampler obtains sampled digital data for a digitalrepresentation of the waveform of the reception signal at the desiredsampling points. The raw data of the digital representation includes theinterference and noise accumulated in the (analog) reception signal,even such impairments that remain if the optional analog filter 411 isused. In general, care must be taken to keep the minimum and maximumamplitudes of the filtered waveforms within the range of theanalog-to-digital converter ADC in the receiver to avoid clipping thesignal. The ratio of the minimum to maximum ranges compared to the meanamplitude for RC and RRC filters tends to be elevated (particularly athigher roll-off factors). Systems relying on super-Gaussian filters, asopposed to RC or RRC filters, in the receiver tend to suffer less fromclipping-related distortion in the ADC, because such systems do not needto accurately capture all of the peak amplitudes of the waveforms.Rather, systems using super-Gaussian filters in the receiver can focuson the intermediate values of the waveform, and can tolerate clipping inthe ADC.

The receiver filter 405 generally represents circuitry for digitallyfiltering the digital representation of the reception signal, e.g., toperform the necessary computations for a Fourier transform of the dataof the digital representation to convert this data from the time domainto the frequency domain, and thereby reject noise frequencies. In orderto optimize or maximize the filtering of the impairments, interferenceor noise from the digital representation of the reception signal, thereceiver filter 405 needs to be optimally matched to the function of thetransmitter filter 404 that shaped the pulses of the transmissionsignal, so that the receiver filter 405 can reverse the function of thetransmitter filter 404, as if in a mirror image. For this reason,conventional practice has been to use the same type of digital filtersat both the receiver and the transmitter. However, the receiver filter405 performs the function known as the higher-order Gaussian, orsuper-Gaussian, e.g., with a formula of:

$\begin{matrix}{{H(f)} = e^{{{- 2^{{2\; N} - 1}} \cdot \log}\; {2 \cdot {\lbrack\frac{f - f_{c}}{f_{3\; {dB}}}\rbrack}^{2\; N}}}} & (3)\end{matrix}$

where f_(3dB) relates to the 3 dB filter bandwidth, f_(c) relates to thefilter center frequency, N is the super-Gaussian order, and f is thefrequency measured in Hertz. In some embodiments, such as in opticalcommunications systems, f_(3dB) is greater than or equal to 1 GHz, or isgreater than or equal to 5 GHz, or is greater than or equal to 10 GHz,or is greater than or equal to 100 GHz, or is from 1 GHz to 1 THz. Insome embodiments, such as in RF systems, f_(c) is greater than or equalto 10 kHz, or is greater than or equal to 100 kHz, or is greater than orequal to 1 MHz, or is greater than or equal to 10 GHz, or is from 1 kHzto 300 GHz. In some embodiments, N is greater than 1, or greater than10, or greater than 50, or from 1 to 100, or from 10 to 50, or from 50to 100.

The optional channel impairments mitigating equalizer and data-retrievalfilter 413 generally represents circuitry for performing any desired orappropriate additional digital filtering of the waveform data of thefrequency domain digital representation. For example, the optionalfilter 413 can be used to remove some channel-induced distortions thatremain in the data at this point. The optional filter 413 is generallyneeded when the channel induces a large amount of distortion on thetransmitter signal.

The de-mapper 414 generally represents circuitry for converting thefiltered waveform data of the digital representation into a stream ofdigital data or serialized bits (e.g., ones and zeroes), or first intothe multi-level symbols and then into the digital data. Ideally, thedigital data at this point would be a reconstruction of the originaltransmission data. However, some error correction is typically performednext. The decoder 415, thus, generally represents circuitry capable ofdecoding the digital data to detect and correct bit-level errors in thedigital data, in accordance with the error correction bits that wereinserted by the ECC encoder 407. The original transmission data has thusbeen retrieved at this point.

In some embodiments, the system can include RC or RRC filters betweenthe sampler 412 and the de-mapper 414 in addition to the receiver filter405. Such pulse shaping in both the transmitter and receiver can, insome cases, be used to satisfy the Nyquist requirement. For example,such RC or RRC filters between the sampler 412 and the de-mapper 414 canreduce ISI.

The output 416 for the retrieved information stream generally representscircuitry for outputting the transmission data. For example, the output416 may provide the transmission data to other components (e.g., aprocessor, electronic memory, etc.) of an overall electronic device ofwhich the receiver 402 is a portion.

As mentioned above, the receiver filter 405 does not specifically matchthe transmitter filter 404. FIGS. 5 and 6 illustrate this mismatch. FIG.5 shows an ideal RC/RRC frequency response graph 500 for differentroll-off factors (e.g., 0.01, 0.1 and 0.5), commonly designated as an αor β parameter. For higher roll-off factor values, the RC/RRC frequencyresponse graph 500 is more curved; but for lower roll-off factor values,the RC/RRC frequency response graph 500 attains more of the ideal squarewave shape representative of square wave data bits. FIG. 6, on the otherhand, shows an ideal super-Gaussian frequency response graph 600 fordifferent exponents N (1-4). For the exponent N=1, Formula 3 (above) isa standard Gaussian function, rather than a super-Gaussian. A moregeneral formulation of a Gaussian function (i.e., the super-Gaussian) isformed by raising the exponent N to any real number (i.e., notnecessarily an integer) greater than 1. For lower values of the exponentN, the super-Gaussian graph 600 is more bell-shaped; but for highervalues of the exponent N (e.g., for N=4 or more), the super-Gaussiangraph 600 attains more of a flat-top and a steeper-edged Gaussianfall-off, similar to the RC/RRC frequency response graph 500 for thelower roll-off factor values. For very large exponents (e.g., greaterthan 10, or greater than 64, or greater than 100) the super-Gaussianfilter can be described as a brick-wall filter, i.e., that completely(or almost completely) cuts off the signal outside of a pre-selectedfrequency.

The RC/RRC frequency response graph 500 is technically the frequencyresponse (frequency domain) “match” for the time domain pulse 301, sincethe transmitter filter 404 performs RC/RRC filtering. Thus, conventionalteaching is that the receiver 402 should use an RC/RRC filter thatmatches (or mirrors the function of) the transmitter (RC/RRC) filter404. In real-world implementations of RC/RRC filters, however, theperfectly symmetrical infinite oscillations of the time domain pulse 301and the steep edges of the RC/RRC frequency response graph 500 cannot beachieved. As a result, it has been discovered in simulations andpractical implementation that the super-Gaussian filter results in amarkedly improved quality in the generation of the digitalrepresentation of the waveform of the reception signal in the receiver402. The improvement is particularly noticeable in implementations usingthe lower roll-off factor values in the RC/RRC filtering of thetransmitter filter 404, but there is still some improvement when usingthe higher values (e.g., 0.1 and higher). In other words, a real-worldimplementation of the super-Gaussian graph 600 is a better match for areal-world implementation of the time domain pulse 301 due to practicalconstraints of the finite resolution of digital-to-analog andanalog-to-digital circuits and/or the truncated response of RC/RRCpulses that are used in real-world communication transmission systems

The performance of communication systems can be characterized using thequality factor (Q-factor), the bit error rate (BER), or the bit errorratio (also BER). In the case of the additive white Gaussiannoise-affected performance, the Q-factor is generally related to the biterror ratio (BER), for example, by a formula such as:

$\begin{matrix}{{{BER} = {\frac{1}{2}{{erfc}\left( \frac{Q}{\sqrt{2}} \right)}}},} & (4)\end{matrix}$

where Q is the quality factor, and erfc is the error functioncomplement. In digital transmission, the number of bit errors is thenumber of received bits of a data stream over a communication channelthat have been altered due to noise, interference, distortion, bitsynchronization errors, etc. The bit error rate is the number of biterrors per unit time. The bit error ratio is the number of bit errorsdivided by the total number of transferred bits during a studied timeinterval. Bit error ratio is a unitless performance measure, oftenexpressed as a percentage.

For a transmission signal, as the signal-to-noise ratio (SNR) improves,it should be expected that the bit error ratio is reduced. In accordancewith the error function complement, as the bit error ratio is reducedthe quality factor is generally increased. Therefore, as thesignal-to-noise ratio increases, the quality factor should alsoincrease. FIG. 7 shows several graphs of Q-factor vs. signal-to-noiseratio (SNR) for several example transmission systems that illustrateimprovements of the transmission system 400 over prior art transmissionsystems. The graphs of Q-factor vs. SNR were generated by simulations ofthe different example transmission systems. The signals for each of thesimulations shown in FIG. 7 were 16-QAM signals.

For the graphs in FIG. 7, the example transmission systems use 1) RRCfilters in both the transmitter and receiver (RRC-RRC graphs 701-703),2) an RC filter in the transmitter and a super-Gaussian (SG) filter inthe receiver (RC-SG graphs 704-706), and 3) an RC filter in thetransmitter and a simple Gaussian (G) filter in the receiver (RC-Ggraphs 707-709). The roll-off-factors for the RC and RRC filters in thesimulations shown in FIG. 7 were 0.01. The excess bandwidth of the SGfilters in the simulations shown in FIG. 7 was also 0.01. Additionally,the top graphs 701, 704 and 707 were generated for transmission systemsin which the receiver samples the reception signal at 4 sampling points,with a sampling rate of 2 samples per symbol; the middle graphs 702, 705and 708 were generated for transmission systems in which the receiversamples the reception signal at 16 sampling points, with a sampling rateof 2 samples per symbol; and the bottom graphs 703, 706 and 709 weregenerated for transmission systems in which the receiver samples thereception signal at 32 sampling points, with a sampling rate of 2samples per symbol. The sampling points in each of these cases wereequally spaced in time. The term “sampling points” in the data shown inFIG. 7 refers to the total number of points used to generate the sampledsignal data (e.g., in an FIR filter). All of the results shown in FIG. 7utilize 2 samples per symbol, although a smaller, and even fractional(i.e. non-integer) sampling rate with respect to the symbol durationinterval can be used in practice, with a similar effect.

The RC-SG graphs 704-706 illustrate the performance of an exampleimplementation of the improved transmission system 400 (FIG. 4), sincethese graphs were generated using an RC filter in the transmitter and asuper-Gaussian (SG) filter in the receiver. Each of the RC-SG graphs704-706 clearly shows the quality factor increasing at almost the samerate as the signal-to-noise ratio increases, even for the exampleimplementation (graph 704) that sampled the reception signal at only 4sampling points.

The RRC-RRC graphs 701-703, on the other hand, illustrate theperformance of an example transmission system in accordance withconventional teaching, i.e., in which the transmitter and receiverfilters are matched RRC filters. In spite of the matched filters,however, the RRC-RRC graphs 701-703 show that the example conventionaltransmission system does not achieve the same level for the qualityfactor as the example improved transmission system 400 does, unless ituses a large number of sampling points, e.g., 32 sampling points (graph703) and with a sampling rate of 2 samples per symbol (interval). Infact, for the example conventional transmission system that uses only 4sampling points (graph 701), the quality factor significantly divergesfrom that of the example improved transmission system 400 as thesignal-to-noise ratio increases, i.e., the performance quality does notcontinue to increase with the signal-to-noise ratio. Additionally, withas many as 16 sampling points (graph 702), the example conventionaltransmission system still does not perform with the same quality as theexample improved transmission system 400 does. Therefore, with fewersampling points, the example improved transmission system 400 is shownto perform significantly better than the example conventionalmatched-filter transmission system does. In particular, with a maximumof just 4-16 sampling points to generate the sampled digital data, theexample improved transmission system 400 can achieve a performance levelfor which the example conventional matched-filter transmission systemneeds at least 32 sampling points.

By way of comparison, the RC-G graphs 707-709 illustrate an exampletransmission system with a different mismatch between the transmitterand receiver filters, i.e., an RC filter in the transmitter and simpleGaussian filter (exponent N=1) in the receiver. The RC-G graphs 707-709show that this example mismatched-filter transmission system fails toachieve the quality factor that the example improved transmission system400 does, i.e., the quality factor increases at a lower rate. In fact,the RC-graphs 707-709 barely change with increasing numbers of samplingpoints. Therefore, the example improved transmission system 400 (withthe higher-order super-Gaussian filter, e.g., N=4 or more) is shown toperform significantly better than the example mismatched-filtertransmission system does.

Although the RRC-RRC graph 703 shows that the example conventionalmatched-filter transmission system with 32 sampling points performs aswell as the example improved transmission system 400 does with 4, 16 or32 sampling points, the requirement for the larger number of samplingpoints is a significant disadvantage for the example conventionalmatched-filter transmission system. The larger number of sampling pointsmeans that more data is generated for the digital representation of thereception signal, which requires a more complex circuitry to generate,filter, equalize, de-map and otherwise process or analyze the data inthe receiver 402 and more time to perform these functions. However,since the example improved transmission system 400 can achieve the samequality results using fewer sampling points, the overall circuitry forthese functions of the receiver 402 of the example improved transmissionsystem 400 can be simpler, smaller, cheaper and faster than that of theexample conventional matched-filter transmission system. Additionally,in some embodiments, since the receiver 402 is capable of producing arelatively high-quality result with relatively few sampling points, thetransmitter 401 need not generate the transmission signal with a veryhigh quality. Since the number of filter taps can be reduced, thetransmitter filter 404 and the arbitrary waveform generator 410 can useless power than that of the example conventional transmission systemdesigned on the theoretical premise of the matched-filter. Additionally,the sampler 412 can be lower quality (e.g., be an ADC with lowerresolution, or a lower number of bits) than a sampler in a system usingconventional matched filters in the transmitter and receiver.

In some embodiments, the super-Gaussian filter is applied in thespectral domain, although two Fourier transforms would need to beperformed on the reception signal. In some embodiments, thesuper-Gaussian filter is applied in the time domain by means of finiteimpulse response (FIR) filters, or tap-delay line filters. In the lattercase, the filter taps are obtained by inverse Fourier transforming thespectral representation of the super-Gaussian. In some embodiments, inthe application of the filter shape, slight deviations from thefunctional form of the super-Gaussian expression can be used: e.g.piece-wise approximations of the super-Gaussian function can be formedfrom formula 3 (above) in the spectral domain. Then the approximationscan be inverse-Fourier transformed to obtain the tap point samples of afinite impulse response (FIR) filter. In this manner, a flat-top in themiddle of the filtered shape is combined with various (i.e. arbitrary)roll-off of the edges, from purely linear (e.g., overall trapezoidshape) to an appropriate polynomial, or similar approximations of theroll-off on the edges of the signal band. In some embodiments, thefilter can be applied purely in the spectral (i.e. Fourier) domain, ifallowed by the associated complexity and/or power dissipation of thatapproach. In some embodiments, the super-Gaussian filter could beapplied by some utilization of wavelet transforms. In some embodiments,the super-Gaussian filter can be adjusted, e.g., to make up for anon-ideal shape of the analog components in the system (such as theoptional analog filter 411) or any deficiencies of the components in thereceiver 402 as a whole, in order to arrive at theoverall-super-Gaussian shape.

In some embodiments, additional signal shaping can occur in thetransmitter and also be corrected for in the receiver, without deviatingfrom the disclosed concepts of using RC or RRC filters in thetransmitter and super-Gaussian filters in the receiver. For example,referring back to FIG. 4, the RC or RRC filter 404 can be used in thetransmitter, and an additional filter can shape the signal using afunction G(ω). The G(ω) shape can be induced at any point between theECC encoder 407 and the waveform generator 410. This additional signalshaping can then be corrected for in the receiver using a filter with aninverse function 1/G(ω). In some cases, the function 1/G(ω) compensatesfor a departure of the transmitter and receiver overall transfercharacteristic from a Nyquist condition for ISI-free communication. Thisshaping correction in the receiver can occur at any point between thesampler 412 and the de-mapper 414. In other words, “imperfect” signalshaping (i.e., non-RC, or non-RRC filter) can be used in the transmitterand be corrected for in the receiver, without detrimentally impactingthe advantages of using the mismatched filters with super-Gaussianfilters in the receiver described herein.

In some embodiments, super-Gaussian filters can be used in thetransmitter and the receiver, as long as the Nyquist requirements aremet to mitigate ISI, i.e. in a combination of an appropriate overallresponse adjusting filter G(ω), as explained in the previous paragraph.For example, a system can include a super-Gaussian filter in thetransmitter, and a super-Gaussian filter in the receiver, and theNyquist requirement can be satisfied by using a G(ω) filter in thetransmitter and/or receiver, adjusting the overall system response tothat of the Nyquist criterion.

While the specification has been described in detail with respect tospecific embodiments of the invention, it will be appreciated that thoseskilled in the art, upon attaining an understanding of the foregoing,may readily conceive of alterations to, variations of, and equivalentsto these embodiments. Any of the method steps discussed above can beconducted by a processing unit, or application specific integratedcircuit with a readable non-transitory medium storing instructionsand/or data for those method steps. The readable medium may be memorywithin an electronic device itself or a network accessible memory. Theseand other modifications and variations to the present invention may bepracticed by those skilled in the art, without departing from the scopeof the present invention, which is more particularly set forth in theappended claims.

What is claimed is:
 1. A method comprising: receiving, by a transmitter, transmission data; generating, by the transmitter, a transmission signal from the transmission data, the transmission signal having a series of shaped pulses; transmitting, by the transmitter, the transmission signal; receiving, by a receiver, a reception signal based on the transmission signal having passed through a transmission channel; sampling, by the receiver, the reception signal to generate sampled digital data; and filtering, by the receiver, the sampled digital data through a receiver filter to regenerate the transmission data wherein: the transmitter generates the shaped pulses using a pulse shaping filter; and the pulse shaping filter and the receiver filter are mismatched.
 2. The method of claim 1, wherein: the receiver filter is a super-Gaussian filter.
 3. The method of claim 2, wherein: the pulse shaping filter is selected from the group consisting of raised-cosine filter and root-raised-cosine filter families.
 4. The method of claim 2, wherein: the super-Gaussian filter in the receiver has excess bandwidth compared to the pulse shaping filter in the transmitter.
 5. The method of claim 2, wherein: the super-Gaussian filter operates with 4-16 sampling points to generate the sampled digital data.
 6. The method of claim 2, wherein: the super-Gaussian filter operates with a maximum of 4 sampling points to generate the sampled digital data.
 7. The method of claim 2, wherein: the super-Gaussian filter operates according to a super-Gaussian function having an exponent of 4 or more.
 8. The method of claim 2, wherein: the transmission signal is an optical signal; and the transmission channel is a fiber-optic channel.
 9. The method of claim 2, wherein: the transmission signal is an RF signal; and the transmission channel is air.
 10. The method of claim 2, wherein: the transmitter further generates the shaped pulses using a second pulse shaping filter that shapes the pulses operating according to a function G(ω); and the filtering at the receiver further comprises filtering the sampled digital data using a filter operating according to a function 1/G(ω).
 11. A transmission system comprising: a transmitter having a pulse shaping filter with which the transmitter generates a transmission signal with a series of shaped pulses for transmission, the series of shaped pulses being generated from digital transmission data; and a receiver having a receiver filter with which the receiver regenerates the digital transmission data from sampled digital data of a reception signal based on the transmission signal having passed through a transmission channel; wherein the pulse shaping filter and the receiver filter are mismatched.
 12. The transmission system of claim 11, wherein: the receiver filter is a super-Gaussian filter.
 13. The transmission system of claim 12, wherein: the pulse shaping filter is selected from the group consisting of raised-cosine filter, or a root-raised-cosine filter.
 14. The transmission system of claim 12, wherein: the super-Gaussian filter in the receiver has excess bandwidth compared to the pulse shaping filter in the transmitter.
 15. The transmission system of claim 12, wherein: the super-Gaussian filter operates with 4-16 sampling points to generate the sampled digital data.
 16. The transmission system of claim 12, wherein: the super-Gaussian filter operates with a maximum of 4 sampling points to generate the sampled digital data.
 17. The transmission system of claim 12, wherein: the super-Gaussian filter operates according to a super-Gaussian function having an exponent of 4 or more.
 18. The transmission system of claim 12, wherein: the transmission signal is an optical signal; and the transmission channel is an optical channel.
 19. The transmission system of claim 12, wherein: the transmission signal is an RF signal; and the transmission channel is air.
 20. The transmission system of claim 12, wherein: the transmitter further comprises a second pulse shaping filter, wherein the second pulse shaping filter shapes the pulses operating according to a function G(ω); and the receiver further comprises a filter operating according to a function 1/G(ω) which compensates for a departure of the transmitter and receiver overall transfer characteristic from a Nyquist condition for ISI-free communication. 